In three dimensions, the vector transformation rule is written as
where .
In two dimensions, this transformation rule is the familiar
In matrix form,
Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.
In three dimensions, the second-order tensor transformation rule is written as
where .
The Cauchy stress is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as
In matrix form,
Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.
Let construct an orthonormal basis of the second order tensor projected in the first order tensor
The stress and strain tensors are now defined by :
and
Then once constructs the bound matrix in the orthonormal base
with
the rotation matrix in base.
is the rotation along the axis in the : base
The associated rotation in the base is :
The rotation of a second order tensor is now defined by :
The élasticity tensor in the : is defined in the : by
and is rotated by:
Introduction to Elasticity